EXTENSIONS of ORDERED GROUPS and SEQUENCE COMPLETION(I) by CHARLES HOLLAND^)

EXTENSIONS OF ORDERED GROUPS AND SEQUENCE COMPLETION(i) BY CHARLES HOLLAND^)

1. Introduction. In this paper the relationships between certain extensions of totally (linearly) ordered groups (o-groups) are studied. The additive notation is employed although the groups are not, in general, assumed to be abelian. The four types of extensions considered are defined as follows : Let G S H be o-groups. (a) H is an a-extension of G if for every 0 < he H there exists ge G and a positive integer n such that h ^ g ^ nh. (b) H is a b-extension of G if for every 0 < n e H there exists g e G and a positive integer n such that g ^ n :£ ng. (c) H is a c-extension of G if for every 0 < h e Ü there exists g e G such that for all integers n, n(h —g) < h. (t) H is a t-extension of G if for every h,h',h"eH with h < h' < h" there exists g e G such that h < g < h". If xe {a, b,c,f}, G is x-closed if G has no proper x-extensions. The connect- ing notion between the different types of extensions is that of sequences in o- groups; of particular importance are cauchy sequences (definition in §2) and pseudo sequences (definition in §3). Several authors have discussed similar con- cepts, but in different or more special cases. Cohen and Goffman [3 ; 4] consider cauchy sequences in abelian groups; Everett and Ulam [8] consider countable cauchy sequences; Banaschewski [1] deals with cauchy filters in partially ordered groups; Gravett [9] relates pseudo sequences to c-extensions in divisible abelian groups. §2 contains definitions and a basic lemma concerning sequences. It is also shown that every r-extension and every c-extension is a ¿-extension, and that every b-extension is an a-extension. Some results concerning r-extensions and cauchy sequences ("j" theorems) are stated without proof. In §3 some theorems ("C" theorems) concerning pseudo sequences and c-extensions, including analogues to the "T" theorems, are proved. In §4 are stated the "B" theorems concerning sequences and b-extensions with proofs only sketched, as the proofs

Received by the editors May 14, 1962. i1) This paper is based on material in the author's Ph.D. thesis, written under the direc- tion of P. F. Conrad, to whom the author expresses sincerest gratitude. (2) The author holds a NATO Postdoctoral Fellowship at the University of Tübingen. 71

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are mostly similar to those in §3. In §6 are examples which illustrate the limita- tion of the theory. The system of numbering is such that Theorems Ti, Ci, and Bi are analogues. 2. Cauchy sequences and i-extensions. Throughout this paper G will denote an o-group. The pairs (Gy,Gy) of convex subgroups of G such that Gy covers Gy are called convex pairs. Gy is normal in Gy and Gy/Gy is o-isomorphic to a subgroup of R, the additive group of real numbers. The groups Gy/Gy are called the components of G. If A and B are subsets of G, A < B means every element of A is less than every element of B. For g e G, | g | is the max of g and —g. A sequence (in G) is a collection {sx | a e A} of elements of G such that A is a well-ordered set with no final element. The breadth of a sequence {sx} is the inter- section of all convex subgroups C of G satisfying: (1) C is covered by a convex subgroup C. (2) For each geC \C there exists peA such that if a, ß > p then sx — sßeC and sx- sß + C <\g\ + C Note that the breadth is a convex subgroup. Here and in several places later on the definition is "one-sided" due to lack of commutativity. This would more appropriately be called "left breadth," the "right breadth" being defined sim- ilarly. The introduction of this complexity seems, however, to be of no particular aid. An element s of G is a limit of the sequence {sa} if for every convex pair (C,C) such that C contains the breadth of {sx} and for every geC \C, there exists v e A such that if a > v, sx — seC and \sx — s\ + C <\g\ + C. We say {sx} converges to s. Lemma 1. If r and s are limits of {sx} then r —s lies in the breadth of {sx}. Proof. By way of contradiction, if r —s is not in the breadth B then r-seC'\C where (C,C) is a convex pair and BçzC. Now for all sufficiently large a, 2\sx — r\ + C<\r — s\ + C. For C'/C is o-isomorphic to a subgroup of the real numbers, and if C'/C is not discrete then there exists g e G such that 2|g| + C<|r — s| + C and by assumption, for all a large enough, Is,, — r\ + C < \g\ + C. If C'/C is discrete it is cyclic generated by some f + C; then for all a large enough | sx - r | + C < \f | + C so that sx — reC, in which case 2\sx- r\ + C = C <\r - s\ + C. Choose a so that sx-r, sx — seC and 2\ sx - r\+ C <\r - s\ + C and 2\ sx - s\ + C <\ r - s\ + C. Then \r-s\ + C = \r-sa + sx-s\ + C ^ \sa-r\ + \s„-s\+C <\r - s\ + C,& contradiction. Hence r —seB. Lemma 2. Let H be an a-extension of G. If {gx} is a sequence of elements of G, if N is the breadth of {gx} in G, and if B is the breadth of {gx}in H, then BnG = N. If geG is a limit of {gx} in G then g is a limit of {gx} in H. Proof. (K,K') is a convex pair of if if and only if (K n G, K' C\ G) is a con-

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vex pair of G. In the definition of breadth, it is required that the differences gx — gß become arbitrarily small in certain components K'/K and K' n G/K n G, respectively. As K' n G/K nG Z K'/K are subgroups of the real numbers, the differences in question become small in K' nG/XnG if and only if they be- beco me small in K'/K. A similar remark holds for the differences gx — g. A sequence {sx} is a left cauchy sequence if its breadth is 0. {sx} is a cauchy sequence if both {sx} and { —sa} are left cauchy sequences. Example 1 of §6 contains a left cauchy sequence which is not a cauchy sequence. In the topology generated by the open intervals, every o-group is a topological group. It is easily seen from the definition that if is a f-extension of G if and only if G is dense in H in the order topology. Banaschewski [1] has shown that the topological completion t(G) of an o-group G is an o-group which extends the order of G. Banaschewski's proof, as well as the earlier and less general ones of Cohen and Goffman [3] and Everett and Ulam [8], proceeds by completing G via "dedekind cuts." It is also possible to construct t(G) by choosing cauchy sequences of appropriate length, showing that they form an o-group under term- wise addition, and continuing in a manner analogous to the cauchy sequence construction of the real numbers. The proof of this, from which the following results easily follow, is notationally involved and sufficiently similar to the classical results that the details are omitted here. Theorem Tl. If H is a t-extension of G and heH\G then there is a cauchy sequence of elements of G with limit h (and with no limits in G). Corollary T2. If every cauchy sequence in G has a limit then G is t-closed. Let T be an ordered set, and for each y e T let Ry be a subgroup of the real numbers. The group H(T,Ry) consisting of those functions in the large direct sum of the Ry whose support is inversely well ordered, is called a Hahn group, and is ordered by calling a function positive if it is positive on the largest element of its support.

Lemma T4. If H(T,Ry) is a Hahn group and either T has no smallest element or y0 is the smallest element of V and Ryo is the reals or the integers, then H(T,Ry) is t-closed.

Proof. Let {fx} be a cauchy sequence in H. It follows from the definition of cauchy sequence that for each y e T except the smallest element of T, if any, for all sufficiently large a and ß, fx(y) =fß(y) =f(y), say. Moreover, if T has a smallest element y0 then the sequence {fx(y0)} is a cauchy sequence in the ordinary sense in R, and hence converges to/(y0), say. It is easy to verify that the function / is an element of H and the limit of {/„}. Theorem T6. G is t-closed if and only if every cauchy sequence in G converges.

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Lemma 3. Every t-extension of G is a c-extension of G if and only if G is t-closed or G has no convex subgroup covering 0. Proof. If G has no convex subgroup covering 0 and if if is a f-extension of G, let 0 < h e K' \K where (K,K') is a convex pair of H and h > 0. Then K ^ 0 and hence the interval h + K of if contains an element g e G (since G is dense in H). Therefore h - geK; that is, n(h —g) < h for all n. Conversely, if every i-extension is a c-extension and if G has a convex subgroup C covering 0, then from the remark at the beginning of the proof of Lemma 2 and the fact (which will be proved presently) that every i-extension is an a- extension, it follows that t(G) has a convex subgroup C covering 0. As t(G) is by assumption a c-extension of G, for every 0

Lemma 4. Every t-extension is a b-extension.

Proof. Let if be a nontrivial i-extension of G and let 0 < he H \G. Then there exists h'eH with h'

Lemma 5. Every c-extension is a b-extension.

Proof. Let if be a c-extension of G and let 0 < h e H. Then h — g = a for some geG where for all n, na < h. Also a —f = b for some/e G where nb < \ a j for all n. Let x = — |/| +/+ geG. It is easily verified that x

Lemma 6. Every b-extension is an a-extension.

Proof. Let if be a o-extension of G. Let 0 < h e H. There exists geG such that g-íLhfLng. Therefore h ^ng :£ nh.

3. Pseudo sequences and c-extensions. For every O^geG there exists a unique convex pair (Gx, G*) such that geGx\Gx. The mapping V: G\{0} -> T(G), where T(G) is the set of all convex pairs of G, defined by V(g) = (Gx, G") is the natural valuation of G. Theorems Cl, C2, C5, and C6 which follow are general- izations of the classical valuation theory (Schilling [11]), already generalized by Gravett [9] to vector spaces (and hence to divisible abelian o-groups). A sequence {sx} is called a pseudo sequence if for all a < ß < y, V(sß — sy) < V(sx — sß). It follows that for each a there exists Xx e T(G)

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such that for all ß > a, V(sx — sß) = Xa. Moreover, for each limit s of {sa}, V(sx — s) = Xx. The breadth of {sj is the same as that defined in the classical theory.

Theorem Cl. If H is a c-extension of G and heH\G then there is a pseudo sequence of elements of G with limit h and with no limits in G.

Proof. The empty set is a pseudo sequence with limit n. By Zorn's lemma there is a pseudo sequence of elements of G with limit h of maximal length (since no element can occur twice in the same pseudo sequence). Let {gx} be a pseudo sequence of elements of G with limit n of maximal length. If {gx} has a limit geG then by Lemma 1, g—h is in the breadth of {gx}. Hence V(g—h) < Xx for all a. Let g(1) = g. Define g(l) by induction for i = 2,3, ••• as follows: since H is a c-extension of G there exists /(,)e G with h — g(1_1) —/(l) = n(,) where V(h(i)) < V(h - g(i_1)). Let g(i) =/(0 + g0"^. It is easy to verify that the sequence {•••,gx, •••,g(1),g(2), •••} is a pseudo sequence with limit n. This contra- dicts the maximality of {gx} and shows that {gx} has no limits in G.

Corollary C2. // every pseudo sequence in G converges, then G is c-closed.

The converse of Corollary C2 is not true, as is shown by Example 2 of §6. For the abelian case, however, Lemma C5, states the converse. The following theorem was proved for abelian G by Ribenboim [10].

Theorem C3. If for every noncovered convex subgroup C of G, C is normal and G/C is t-closed, then G is c-closed.

Proof. Suppose that H is a proper c-extension of G and that heH\G. By Theorem Cl there is a pseudo sequence {gx} with limit h and no limits in G. Let B be the breadth of {gx} in G. Then B is noncovered and hence normal in G. The first aim is to show that {gx + B} is a cauchy sequence in G/B. As before, for ß > a let V(gx - gß) = Xx = (C'x,C"), where C'x =CxnG, C" = C*r\G, and (Cx, C) is a convex pair in H. Then B = (f]xC)nG. Denote by 4>xthe inner automorphism of H induced by gx, and by \p the inner automorphism of H induced by h. Since gx — heC, and for ß > a, gx — gßeC, Cx oc, Cß c C and hence Cß(pß c C"4>„= Cx4>x.Hence (f)xCx) n G is a noncovered convex subgroup of G, and by assumption is normal in G. In particular, ((C\xC4>x)r\G)=n«c"4vFor each ß> c^ß=rv^a=cn.c'w»-Hence c =5n«c^-From thisfo,iows ri/»c' ^ n«c^ ■ n«c^- Since -h + gx = - n + (ga - n) + n, then - h + gxeC>. Hence — ia + 8ß— —gx + h — h + gße(Cil/)u(Cßtl/). From this and the last statement of the preceding paragraph it follows that {gx + B} is a cauchy sequence in G/B.

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By assumption {gx + B} converges to some g + B. Then g is a limit of {gx}, a contradiction.

Lemma C4. In any Hahn group, all pseudo sequences converge. Thus all Hahn groups are c-closed. Proof. If if is a Hahn group and {hx} is a pseudo sequence of breadth B, then {hx + B} is a cauchy sequence in H/B, which is also a Hahn group. Thus by Lemma T4 and Theorem T6 {hx + B} has a limit h + B. Hence ft is a limit of {hx}. The last half of the lemma follows from Corollary C2. Lemma C5. If G is abelian and c-closed, then every pseudo sequence in G converges.

Proof. Suppose that {sx} is a pseudo sequence in G with no limits in G. For each a and all ß > a, V(sx — sß) = Xx, say. Note that for each integer n ^ 0, {nsx} is a pseudo sequence. If {nsx} has g as a limit and if {msx} has h as a limit then it is easily verified that {(n + m)sx) has g + h as a. limit. Thus the set of all integers p such that {psx} has a limit in G is a subgroup of the integers, generated by some n0. It is also easily seen that all {psx} are pseudo sequences in any extension of G. By Hahn's theorem G can be embedded in a divisible Hahn group H. By Lemma C4 any pseudo sequence in if converges. Choose seif as follows: if {nsx} has no limit in G for every integer «#0, let s be any limit of {sx} in if. If {nsx} has a limit in G for some integer n, let the group of such integers be generated by n0, let s' be a limit of {n0sx} in G and let s = s'/n0. Then in either case s is a limit of {sx}. Now G is properly contained in G', the subgroup of H generated by G and s. It will now be shown that G' is a c-extension of G. Let B = {V(h) | h is in the breadth of {sx}}. That is, B is those elements of T(if) which are less than every Xx. Let g + nse G', where geG. Case 1. V(g + ns) = y$B. Then there exists ß such that V(ns — nsß)

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Proof. Since G has no smallest convex subgroup, there exists a sequence {6X}of elements of G such that for a>ß, V(6X)< V(9ß), 0 is the greatest lower bound of {dx}, and for each or, 6x+s e G (since G is dense in t(G)). Let rx = 9x + s. Then for ß > a, V(r„ — rß) = V(6X)= V(rx - s). Hence {rj is a pseudo sequence with limit s. The sequence { —s + 6X+ s} also has greatest lower bound 0. Since V( —rß + rx) = V( —s + (s - rß) + s — s + (rx — s) + s) which is less than or equal to the greater of V( —s + 6X+ s) and V( —s + 9ß + s), {ra} is also a cauchy sequence.

Lemma C7. If G is abelian and c-closed, then for every noncovered convex subgroup C of G, G/C is t-closed.

Proof. For a noncovered convex subgroup C, G/C has no convex subgroup covering 0. If G/C is not i-closed, then by Theorem T6 there is a nonconvergent cauchy sequence {sx + C} in G/C. By Lemma 7 there is a nonconvergent pseudo sequence {rx + C} in G/C. Then {ra} is a nonconvergent pseudo sequence in G. Hence by Theorem C6, G is not c-closed.

Theorem C8. If G is abelian, then G is c-closed if and only if G/C is t-closed for every noncovered convex subgroup C of G.

Proof. Lemma C7 and Theorem C3.

4. Sequences and ¿-extensions.

Theorem Bl. If H is a b-extension of G and heH\G then there is a sequence of elements of G with limit h and with no limits in G. Proof. The proof proceeds as far as possible along the lines of and with the notation of Theorem Cl. Let {gx} be a pseudo sequence of elements of G with imit h and of maximal length. If {gx} has no limits in G, the proof is complete. If {gx} has a limit g in G, let g(1) = g and define g(,)as before, as far as possible. It cannot happen that for every /'= 2,3, •••,g(0exists,forthen{"-,ga, •••,g(I), g(2),---} is a pseudo sequence with limit h and longer than the maximal such sequence {gx}. Hence for some i0 with h — g(io)e C \C where (C,C) is a convex pair, there is no /(io+1) e G such that/(io+1) + C = h - giio) + C. However, in this case it follows from the definition of ¿-extension that C nG/CC\G is not discrete. Thus there is a cauchy sequence (in the classical sense) {e(i) + C} in C'/C with h — g(,o) + C asalimit, and with e(i)e G. Then the sequence {•■•,gx, —,ga\g(2), •••,g(io),e(1) + g(io\e(2)+g(io),---} has h as a limit and no limits in G.

Corollary B2. If every sequence in G has a limit then G is b-closed.

Theorem B3. If for every convex subgroup C of G, C is normal and G/C is t-closed, then G is b-closed.

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Proof. If {sx} is a sequence of breadth C in G, and if C is not covered, then in a manner similar to the proof of Theorem C3, it can be shown that {sx + C} is a cauchy sequence in G/C. If C is covered by C, then there is a p such that for all p., sx — sße C. As C'/C is abelian, it is easily verified that for all xeC and for all a, ß > p, — sx + x + sx + C = — sß + x + sß + C. Define an automorphism 4> of C'/C by (x + C) p. Then for all a, ß > p, - sß + sx+ C = (sx- sß + C)~1; that is, -sß + sx < g + C. Hence {sx + C} is a cauchy sequence in G/C. The rest of the proof is like that of Theorem C3. Lemma B4. If H is a Hahn group in which every component is the real numbers or the integers, then all sequences in H converge. The proof follows quickly from Lemma T4.

Lemma B5. // G is abelian and b-closed then every sequence in G converges. Proof. G can be embedded in a Hahn group H in which every component is the real numbers. In H every sequence converges by Lemma B4. The rest of the proof follows the model of the proof of Lemma C5.

Theorem B6. If G is abelian, G is b-closed if and only if every sequence in G converges. Proof. Corollary B2 and Lemma B5. The following is also easily proved.

Theorem B8. If G is abelian, G is b-closed if and only if for every convex subgroup C, G/C is t-closed.^) Ribenboim [10] calls an abelian o-group G algebraically complete if G/C is r-closed for all convex subgroups C. He shows that such a group is c-closed (which follows here Theorem B8 and the fact that every c-extension is a b-extension), and that every abelian o-group canjbe "algebraically completed." This follows from some remarks in the next section. 5. Remarks on closure. Every o-group G has a r-closed i-extension, namely the group t(G) discussed in §2. Every o-group has an a-closed a-extension and a c-closed c-extension, and the cardinality of all a-extensions of a given group is limited (Conrad [6]). Since every b-extension is an a-extension (Lemma 6), it follows from Zorn's lemma that every o-group has a b-closed b-extension.

(3)By way of comparison, Cohen and Goffman [4] prove: An abelian o-group G is a-closed if and only if for every convex subgroup C (# G), G/C is /-closed and nondiscrete.

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The proof of Lemma C5 shows that if G is abelian and has no proper abelian c-extensions then every pseudo sequence in G converges. Hence by Corollary C2, every such G is c-closed. Likewise from Lemma B5, if G is abelian and has no proper abelian b-extensions then every sequence in G converges, so that every such G is b-closed by Corollary B2. Thus by Zorn's lemma, for any abelian G there is a maximal abelian b(c)-extension, and by the above argument this extension is b(c)-closed. Moreover, it is well known that t(G) is abelian whenever G is, and that the Hahn group H(V,Ry) where T is the set of convex pairs of G and Ry is the real numbers, is an a-closed a-extension of G. Thus : Let xe{t,c,b,a}. G has an x-closed x-extension. If G is abelian, G has an abelian x-closed x-extension. It is well known that the r-closed i-extension is unique to within equivalence; that is, if H and H' are r-closed f-extensions of G, then there is an o-isomorphism of H onto H' which leaves the elements of G fixed. Also, Conrad [5] showed that for an abelian G the abelian a-closed a-extension is unique to within equivalence. Example 3 of the next section shows, however, that it is possible for an abelian G to have two nonequivalent a-closed a-extensions (one of which is non-abelian). The situation is more chaotic for b- and c-closures as is shown in Example 4 which gives an abelian group with two nonequivalent abelian b(c)- closed b(c)-extensions.

6. Examples. Example 1. A left cauchy sequence which is not a cauchy sequence. The group used here was first constructed by Clifford [2] for another purpose (and in a slightly different presentation). G is generated by the symbols g(r) where r is a rational number, and subject to the following condition: if r > s, g(r)g(s) = g(2s — r)g(r). Each word can be put into normal form g = (g(ri))m,(g(r2W2---(g(rn))m" where rt < r2 < ■■•< rn. g is to be positive if and only if m„ > 0. Then the set of convex pairs of G is isomorphic as an ordered set to the rational numbers. One can easily verify the computational laws, for r > s,

g(r)~1g(s) = g((r + s)/2)g(r)-\ and g(r)~^(s)"1 = g((r + s)/2)~1 g(r)~».

Now consider the sequence {s¡} defined by

s, = g(-i)g(-i + l)-g(-l).

For n,m = 1,2, •••, and n^m,

-i Í (g(-m)g(-m + ].)■■■g(-n- 1))_1 if n < m, S"Sm ' 1 g(-n)g(-n + l)-g(-m-l) if n > m.

In any event, V(s„s~l) = -(min{m,n}) - 1. Hence {s¡}is a left cauchy sequence, and even a pseudo sequence. On the other hand, for m,n —2,3, —,and m ^ n,

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,-is | (g(-m)-g(-n- l)s„) 1s„ if«m,

(-g(-n - 1 + 1/2" + 2/2""1 + ••■+ («- l)/22 + n/2)'1) if n < m, (-g(-m - 1 + l/2m + 2/2m_1 + - + (m- l)/22 + m/2)) if n > m,

(-g(l/2n-2)T1iînm.

Hence V(s~1s„) > —2. Thus {s¡} is not a cauchy sequence. The group G provides an example also of an a-closed (and hence c-closed) group with a nonconvergent pseudo sequence. If there were an a-extension of G in which {sj converged, the sequence would necessarily have only one limit, as its breadth is 0. But it can be shown that any sequence with exactly one limit is a cauchy sequence. As the sequence {s¡} is not a cauchy sequence, this shows that {s¡} cannot converge in, for example, an a-closed a-extension of G. The following example gives a much simpler group with the same property. Example 2. An a-closed group with a nonconvergent pseudo sequence. The following is a standard device for constructing o-groups. Let A and B be o- groups and let r be a homomorphism from B to A0(A), the group of order-preserving automorphisms of A. Let S be the set A x B ordered lexicohrapgically from the right and with addition defined by (a,b) + (a', b') = (a(b'r) + a',b + b'). Then S is an o-group, the splitting o-extension of A by B determined by r. Denote S by A xrB(*~) where the arrow indicates the lex order from the right. Let A be the Hahn group Jí(i, R¡) where I is the set of integers and for each i e I, R¡ is the real numbers. Let B' be the Hahn group if (N, R„) where N is the set of negative integers and for each ne N,Rn is the real numbers. Let B be that subgroup of B' consisting of those functions which are "initially an integer": that is, feB if and only iffe B' and there exist integers nf < 0 and kf such that if i < nf then f(i) = kf. Define an o-automorphism y of A as follows: (/y)(0 =f(i — !)• Define r:B-> A0(A) by fr = ykf. Then r is a homomorphism. Let G = A xrB(<-). For each n e N let/„ e B be defined by

(1/2 if n m.

Now consider the pseudo sequence in G, {s¡: / = 1,2, •■•} where s¡ = (0,/_¡). It will now be shown that {s¡} cannot have a limit in any a-extension of G. Supposing, by way of contradiction, that if is an a-extension of G in which {s¡} has a limit s, we have first that the breadth of {sj in G is A x 0 « A. As A is a Hahn group with all components the real numbers, A is a-closed, Let A' = A x 0. A' is a convex subgroup of G. Since H is an a-extension of G, and A'

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is a-closed, then A' is a convex subgroup of H. Again, as H is an a-extension of G, the set of convex subgroups of H is isomorphic as an ordered set to the set of convex subgroups of G. Hence the set of convex subgroups of H which con- tain A' is isomorphic to the negative integers. Any inner automorphism induces an order preserving automorphism of the set of convex subgroups, and therefore A' is normal in H, for otherwise there would be an automorphism of H mapping A' onto a convex subgroup which properly contains A', and hence an order- preserving function mapping the negative integers onto a final segment of them- selves, which is impossible. It can be shown that if {gx} is a cauchy sequence (in any o-group) with limit g, then {2gx} is a cauchy sequence with limit 2g. In H/A', {s¡ + A'} is a cauchy sequence with limit s + A'. Therefore {2s¡ + A'} is a cauchy sequence with limit 2s + A'. Hence 2s is a limit of the sequence {2s,} in H. Another limit of {2s,} is / = (0,(—,1,1, l))eG. Hence by Lemma 1, 2s—fe A'. Let = —2s + x + 2s. Since A' is normal (¡>induces an o-automorphism of A', and since A' is abelian, for any y e A', y

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clearly there is no element of G whose difference with y — x has smaller valuation, since the only thing of smaller valuation is 0 and y - x $ G. This contradiction shows that no such a exists. Note that every component of E and F is either the integers or the real numbers. Thus since E and F are c-closed, they are b-closed. References 1. B. Banaschewski, Über die Vervollständigung geordneter Gruppen, Math. Nachr. 16 (1957), 51-71. 2. A. H. Clifford, A non-commutative ordinally simple linearly ordered group, Proc. Amer. Math. Soc. 2 (1952),902-903. 3. L. W. Cohen and C. Goffman, The topology of ordered abelian groups, Trans. Amer. Math. Soc. 67 (1949),310-319. 4. -, On completeness in the sense of Archimedes, Amer. J. Math. 72 (1950), 747-751. 5. P. F. Conrad, Embedding theorems for abelian groups with valuations, Amer. J. Math. 75 (1953), 1-29. 6. -, On ordered division rings, Proc. Amer. Math. Soc. 5 (1954), 323-328. 7. -, Extensions of ordered groups, Proc. Amer. Math. Soc. 6 (1955), 516-528. 8. C. J. Everett and S. Ulam, On ordered groups, Trans. Amer. Math. Soc. 57 (1945), 208-216. 9. K. A. H. Grave«, Valuedlinear space, Quart. J. Math. Oxford Ser. (2) 6 (1955), 309-315. 10. P. Ribenboim, Sur les groupes totalment ordonnés, Summa Brasil. Math. 4 (1958), 1-64. 11. O. F. G. Schilling, The theory of valuations, Math. Surveys No. 4, Amer. Math. Soc, Providence, R. I., 1950.

Tulane University, New Orleans, Louisiana Universität Tübingen, Tübingen, Germany

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